For compact complex surfaces (i.e., when $\dim X =2$) the map is always surjective. See Theorem 2.10, p.141 in

<cite authors="Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius">_Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius_, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 4. Berlin: Springer (ISBN 3-540-00832-2/hbk). xii, 436&nbsp;p. (2004). [ZBL1036.14016](https://zbmath.org/?q=an:1036.14016).</cite>

In general, the $\partial \bar{\partial}$-lemma (Lemma 13.6 p. 44 of the reference above) says that, if $X$ is a compact complex manifold such that 

 1. the  Frölicher spectral sequence degenerates at $E_1$ and 
 2. there is a formal Hodge decomposition,

then $H^{p, \, q}(X)$ coincides with the subspace of $H^{p+q}(X)$ representable by closed forms of type $(p, \, q)$.